A note on commuting pairs of matrices

We give a short proof of the Motzkin-Taussky result that the variety of commuting pairs of matrices is irreducible. An easy consequence of this is that any two generated commutative subalgebra of n×n matrices has dimension at most n. We also answer an old question of Gerstenhaber by showing that the set of commuting triples of n×n matrices is not irreducible for n≥32.